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Dror Bar-Natan:
Talks:
# Talks at Harvard University

### October 28 and 29, 2004

## Khovanov Homology for Tangles and Cobordisms.

** Colloquium, October 28. **

**Abstract. ** In my talk I will display one complicated picture
and discuss it at length, finding that it's actually quite simple.
Applying a certain 2D TQFT, we will get a homology theory whose Euler
characteristic is the Jones polynomial. Not applying it, very cheaply
we will get an invariant of tangles which is functorial under
cobordisms and an invariant of 2-knots.

Why is it interesting?

- It has several generalizations, but as a whole, we hardly understand
it. It may have significant algebraic and/or physical ramifications. In
fact, it suggests that much of algebra as we know it (or at least quantum
algebra as we know it), is a shadow of some "higher algebra".
- It is a knot/link/tangle invariant stronger than the Jones polynomial,
and an invariant of 2-knots in 4-space.
- It seems stronger than the original "Khovanov Homology".
- It is functorial in the appropriate sense, and Rasmussen (math.GT/0402131)
uses it to do some real topology.

**The picture:**

**Handouts: **
MoreFormulas.pdf.
NewHandout-1.pdf,

**Transparencies: **
Reid2Proof.pdf,
R3Full.pdf,
FrameRack.pdf.

See also my paper Khovanov's
Homology for Tangles and Cobordisms.

##
Trace Groups, Skein Modules and My Problems with Khovanov-Rozansky.

** Gauge Theory and Topology Seminar** (though it's really an
algebra talk)**, October 29 **

**Abstract. ** We will *really* see how the Euler
characteristic of Khovanov homology comes to be the Jones polynomial;
this will take introducing the notion of "trace groups" and
generalizing the notion of "Euler characteristic" to be valued in more
than just integers or power series. I will then quickly explain all
that I know about Khovanov-Rozansky homology which is to HOMFLY like
Khovanov homology is to Jones. "Quickly" and "all that I know"
correctly indicate that I know too little, and in particular, we don't
know how to play the trace groups game in this arena. But even so, the
basic Khovanov-Rozansky story is good and involves a novel technique.
How come it's not in every book?

**Handouts: **
TraceGroups.pdf,
idea.jpg,
KRHDefinitions.png,
KRHDefinitions.nb.

See also my paper Khovanov's
Homology for Tangles and Cobordisms and the paper math.QA/0401268
by Khovanov and Rozansky.